STACK 


JFFEGTIVE  METHODS 

IN 

lECHANIGAL    DRAWING 


EVANS 


EFFECTIVE  METHODS 

IN 

MECHANICAL  DRAWING 

THE 
GEOMETRY  OF  DRAFTING 

AND 
KINKS  AND  SHORT  CUTS 

By  FREDERICK  H.  EVANS,  M.  E. 

Assistant  Professor  of  Manual  Arts,  Bradley  Polytechnic  Institute,  Feoria,  Hi. 

Formerly  Draftsman  for  the  Ironton  Engine  Co.,  Ironton,  O., 

the  Link  Belt  Machinery  Co.  and  Sargent 

and  Lundy,  Chicago. 


THE    MANUAL   ARTS    PRESS 
PEORIA,  ILLINOIS 


COPYRIGHT,     1913, 

FREDERICK   H.   EVANS. 


FOREWORD. 

In  the  text  that  follows  are  presented  geometrical  problems 
that  have  risen  in  the  author's  work  in  machine  drafting, 
covering  a  period  of  over  ten  years,  and  the  solutions  given  are 
the  best  solutions  arrived  at  after  many  hours  of  study.  The 
problems  are  presented  in  their  own  language — in  the  drawing 
board  language  of  conventional  lines.  Mathematical  formulae 
are  reduced  to  a  minimum.  Kinks  and  short  cuts  demonstrate 
the  use  of  paper  as  a  tool  in  such  a  way  as  to  render  the 
mechanical  work  of  solving  the  problems  in  this  subject  much 
more  accurate,  less  difficult,  and,  we  hope,  more  interesting. 

In  making  working  drawings  of  bevel  gears  (especially  when 
the  axes  are  not  at  right  angles)  the  draftsman  finds  himself 
facing  a  difficult  task  in  computation.  A  method  is  here  given 
that  has  been  tested  and  found  to  be  the  simplest,  shortest,  and 
most  accurate  known  to  the  author. 

This  book  is  essentially  a  reprint  of  selected  sections  of  "The 
Drafting  Room  Series",  a  larger  work  by  the  same  author. 


2O65968 


GEOMETRICAL  CONSTRUCTION. 

1.  Machine  parts  are  largely  made  up  of  solids  bounded 
by  plane  and  cylindrical  surfaces.  The  reason  for  this  is  that 
these  surfaces  are  the  easiest  to  construct  and  measure. 

The  chief  uses  of  the  planer,  shaper,  and  milling  machine 
are  to  construct  plane  surfaces,  while  the  drill  press,  boring 
mill,  and  lathe  are  to  construct  cylindrical  surfaces. 

An  object,  the  surfaces  of  which  are  planes,  has  sharp 
corners.  Sharp  corners  are  avoided  in  machine  construction, 
being  either  rounded  off  or  filled  in.  The  rounded  corners 
are  generally  represented  on  drawings  by  circular  arcs  which 
are  tangent  to  straight  lines. 

If  an  object  is  bounded  by  surfaces  other  than  planes  the 
surfaces  must  be  tangent  to  each  other  in  order  to  avoid  sharp 
corners.  For  these  reasons,  methods  of  passing  circles  and 
lines  tangent  to  each  other  are  important  to  draftsmen. 

The  essential  steps  in  drawing  a  circular  arc  tangent  to  two 
lines  are  as  follows:     (See  Plate  2.) 
Determining  the  radius. 
Locating  the  center, 
Locating  the  starting  point  of  the  arc, 
Locating  the  stopping  point  of  the  arc. 

The  starting  and  stopping  points  are  points  of  tangency. 

Tangent  problems  involving  straight  lines  and  circular  arcs 
are  solved  by  an  application  of  the  following  general  rules: 

1.  (a)  If  a  circular  arc  is  tangent  to  a  given  straight 
line,  its  center  must  lie  on  a  line  parallel  to  the  given  line  and 
at  a  distance  from  it  equal  to  the  radius  of  the  circle.  (See 
"Another  Way"  Plate  2.) 

(b)  The  point  of  tangency  must  be  at  the  foot  of  the 
perpendicular  dropped  from  the  center  of  the  arc  to  the  line. 


6  THE  GEOMETRY  OF  DRAFTING 

2.  (a)  If  a  circular  arc  is  tangent  to  a  given  circle,  its 
center  must  lie  on  one  of  two  circles  concentric  with  the  given 
circle;  its  radius  being  equal  to  either  the  radius  of  the  given 
circle  plus  the  radius  of  the  arc,  or,  the  radius  of  the  given 
circle  minus  the  radius  of  the  arc. 

(b)  The  point  of  tangency  must  be  at  the  intersection  of 
the  line  joining  the  two  centers,  with  the  given  circle. 

Solutions  to  a  number  of  typical  problems  are  here  given. 
If  the  student  is  to  enable  himself  to  solve  problems  that  he 
meets  in  practice,  he  must  not  follow  these  solutions  blindly 
but  must  grasp  the  underlying  principles  and  see  the  reasons 
why. 

These  are  not  only  possible  solutions.  The  merit  of  a 
geometrical  solution  consists  in  its  fewness  of  steps  and  its 
avoidance  of  multiplication  of  errors. 

Compare  the  two  methods  on  Plate  9. 

Problems.  Make  drawings  of  the  problems  on  the  succeed- 
ing pages,  following  the  steps  as  given. 


THE  GEOMETRY  OF  DRAFTING 


8 

3 

i 


ra 

1    1 

\o  ^ 

^ 

//• 

re/- 

,   1 

•s 

I  t 

PLATE  1. 


THE  GEOMETRY  OF  DRAFTING 


GIVEN. 

REQUIRED:  A  drawing 
of  The  figure  to  scale. 


4-    - 


r> 


IN  DFfAWlN6  ARCS 


ANOTHER  WAY: 

PLATE  2. 


THE  GEOMETRY  OF  DRAFTING 


6/VEN- 

REQUIRED:  A 
drawing  of  the  figure 
to 


BLOCKING  Our 


L  OCA  TING  LOCA  TING  POINTS 

CENTERS  or  JANGENCY  AND 

0FfAWIN6  3MALL  ARCS. 

FINISH 
PLATE  3. 


10 


THE  GEOMETRY  OF  DRAFTING 


GIVEN 


- 


Slip  4-5°Trianglt 
to  position 


Hold  30*60 
in  This  posi- 
tion 

Draw  1 

Slip  45° and  draw  2,  5lip  45~anddraw  3. 


REQUIRED 

Hnish  H 8cV lines 

with  T  square  and 

triangle's,  and  the 

oblique  lines  by  the 

SET  TRIANGLE  METHOD 

shown 

PLATE  4. 


THE  GEOMETRY  OF  DRAFTING 


PLATE  5. 


12 


THE  GEOMETRY  OF  DRAFTING 


GIVEN 


:  A  drawing 
of  the  figure 


to  A  5. 


BLOCKING  Our,   FINDING 
AND  TANGENT  POINTS,  AND  DRAWING  ARC. 


PLATE  6. 


THE  GEOMETRY  OF  DRAFTING 


13 


LOCATING  CENTERS  LOCATING  TANONT  POINTS 

AND  DRAWING  SMALL  ARC 5 

PLATE  7. 


14 


THE  GEOMETRY  OF  DRAFTING 


GlVCN. 

REQUIRED:  A  drawing 
of  thf  fiqure  fo  zcale. 


BLOCKING  OUT  AH o 
LOCATING  CENTERS. 


LOCATING  TANGENT  POIHTS 
DRAWING  <5MALL  ARCS. 


PLATE  8. 


THE  GEOMETRY  OF  DRAFTING 


15 


GIVEN 


PLATE  9. 


16 


THE  GEOMETRY  OF  DRAFTING 


GEOMETRICAL  CONSTRUCTION 
CIRCLES  A  HP  TANGENTS 

SYMBOLS 

Given  Points  o 

Given  Lines 

Given  Radii  •:•— •- 

Required  Points       o 

Required  Lines 

Required  Radii        o *- 

Construction  Points  o 
Construction  Lines 
Construction  Radii    o —          — «- 


3. 


6. 


PLATE  10. 


THE  GEOMETRY  OF  DRAFTING 


17 


7. 


GEOMETRICAL  CONSTRUCTION 
CIRCLES  ANP  TANGENTS 


4    t 


10. 


THE  GEOMETRY  OF  DRAFTING 


GEOMETRICAL  CONSTRUCTION 
CIRCLES  AND  TANGENTS 


19. 


^0. 


PLATE  12. 


THE  GEOMETRY  OF  DRAFTING  19 


NOTE: — DRAW  TO  SCALE  PLATES  13  AND  14.    AFTER 

MAKING  A  DRAWING  OF  PLATE  16  USE  IT  TO  DRAW  A 
MACHINE  SLIDE,  FOLLOWING  THE  DIRECTIONS  GIVEN  ON 
PLATE  18,  ASSUMING  VALUES  FOR  C1  AND  C2- 


20 


THE  GEOMETRY  OF  DRAFTING 


PLATE  13. 


THE  GEOMETRY  OF  DRAFTING 


21 


PLATE  14. 


22 


THE  GEOMETRY  OF  DRAFTING 


I 


'  I 


/1F----- 
•fH 


tJUsa. 


V? 


lp-1 


PLATE  15. 


THE  GEOMETRY  OF  DRAFTING 


23 


PLATE  16. 


24 


THE  GEOMETRY  OF  DRAFTING 


PLATE  17. 


THE  GEOMETRY  OF  DRAFTING  25 


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26 


KINKS  AND  SHORT  CUTS 

KINKS  AND  SHORT  CUTS. 


2.  The  French  or  Irregular  Curve.  The  French  curve 
is  used  as  a  ruler  for  drawing  non-circular  curves.  When  a 
curve  is  determined  by  a  number  of  points,  a  smooth  freehand 
curve  should  first  be  drawn  lightly  in  pencil  thru  the  points. 
The  French  curve  is  then  fit  to  a  part  of  the  curve  by  trial  and 


FIG.  1. 

a  line  drawn  as  far  as  the  French  curve  follows  the  general 
direction  of  the  curve.  The  French  curve  is  then  moved  to 
another  position  so  that  it  fits  farther  along  the  curve  and  also 
follows  the  part  of  the  curve  just  drawn  for  a  short  distance. 
This  operation  is  repeated  until  the  curve  is  completed. 

The  heavy  line  a,  c,  b,  e,  d,  f,  g,   Fig.   1,  is  being  inked. 
Position  1  of  the  curve  is  drawn  in  light  lines  and  shows  only 


KINKS  AND  SHORT  CUTS 


27 


a  part  of  the  curve.  Position  2  in  heavy  lines  shows  the  entire 
curve,  and  position  3  in  light  lines  shows  only  a  part. 

In  position  1  ab  is  drawn,  in  position  2  cd,  and,  in  position 
3  ef. 

3.  Fig.  2  shows  an  instrument  designed  by  the  author. 
Its  outlines  consist  of  tangent  circular  arcs  of  different  radii. 
The  small  blackened  areas  at  A,  B,  C,  D,  etc.,  are  holes  cut 
thru  the  instrument.  The  corners  of  these  holes  are  centers 
of  the  arcs  a,  b,  c,  d,  etc.  Its  operation  is  as  follows: 


FIG.  2. 

Use  the  instrument  as  a  French  curve  to  pencil  the  curve 
to  be  inked,  marking  the  ends  of  the  arcs  (1,  2,  3,  etc.)  passed 
over  by  the  pencil,  and  also  their  respective  centers  (A,  B,  C, 
etc). 

After  penciling  the  result  is  as  Fig.  3.  The  centers  and 
tangent  points  being  given,  the  curve  can  now  be  inked  with 


28 


KINKS  AND  SHORT  CUTS 


the  compass.  A  trial  will  convince  the  experienced  that  this  is 
a  saver  of  blots  and  a  preventive  of  their  accompanying  losses 
of  time  and  temper.  This  instrument  may  be  made  out  of 
cardboard,  and  by  following  the  solution  of  Prob.  27,  Plate 
12,  its  outline  may  be  made  to  approximate  any  curve. 

4.     Templets  of  Standard  Parts.     The  templet  method 
is  a  means  of  reduplicating  small  drawings.     The  drawing  to 
be   reduplicated    is   made   and    trans- 
fered    to   another   piece   of    paper   by 
pricking  thru  the  essential  parts. 

Fig.  4  shows  a  templet  devised  by 
the  author.  From  this  a  bolt  head 
or  nut  can  be  laid  out  by  pricking 
thru  the  holes  surrounded  by  the 
small  circles.  The  process  is  as  fol- 
lows: 

The  pricker  point  is  placed  at  A, 
s  Fig.   5.      If  it   is  a  bolt  head   to  be 
drawn,  the  notch   H   of  the  templet 
is  slipped  against  the  point;  if  a  nut, 
H   is  used.     The  holes  are  pricked  thru  as 


FIG.  3. 


the  notch  below 
shown. 

Lay  out  lines  and  arcs  are  drawn.     Fig.  6. 

The  finished  head. 
Fig.  7. 

A  templet  when 
made  should  be  filed 
away  for  future  use. 

5.  A  Master 
Templet.  Where 
there  are  a  number  of 
sizes  of  standard  parts 
a  master  templet  can 
be  made  from  which  different  sizes  can  be  laid  off  direct  or 
from  which  individual  templets  can  be  made.  Fig.  8  shows 
such  a  templet  of  a  nut. 


o       o 

o       o 

S"~    ^V 

O 

U.  J.  5t'd. 
Unch 

o 

/^"=^ 

HE 

40 

3? 

fH 

NUT 

IJf 

FIG.   4. 


KINKS  AND  SHORT  CUTS 


29 


In  making  this  templet,  the  smallest  and  the  largest  standard 
size  nut  was  accurately  laid  out  to  scale  from  a  table  of 
dimensions.  Lines  were  then  drawn  connecting  the  principal 
points  of  each. 

To  lay  out  a  nut  of  any  other  size,  the  process  is  as  follows : 

From  A  and  B  on  the  line  AC  lay  off  the  bolt  diameter  of 

the  required  nut  and  draw  the  lines  FG  and  DE.     DE  marks 

I 


FIG.    5.  FIG.    6.  FIG.   7. 

the  top  of  the  nut,  and  the  intersections  of  FG  with  the  lines 
connecting  the  corner  layout  points  of  the  largest  and  smallest 
sizes,  locates  the  corner  layout  points  for  this  particular  one. 


FIG 


A  vertical  line  dropped  to  the  line  K  from  the  intersection 
of  FG  with  the  line  H  locates  the  center  of  a  small  arc.  The 
center  of  the  large  arc  can  be  found  by  the  intersection  of  a  45° 
line,  drawn  from  an  outside  corner,  with  the  center  line. 

By  laying  off  the  proper  height  this  same  templet  may  be 
used  for  a  bolt  head. 


30 


KINKS  AND  SHORT  CUTS 


6.  Section  Lining.  In  section  lining  or  "cross  hatching" 
the  spacing  is  generally  done  by  eye.  Where  a  great  deal  of 
hatching  is  done  it  saves  the  eyes  to  use  a  mechanical  device  to 
do  the  spacing.  There  are  a  number  of  section  liners  on  the 
market,  but  most  of  them  have  other  objectionable  features 
besides  the  price. 


FIG.   9. 

Fig.  9  shows  a  common  and  simple  device  for  mechanical 
spacing.  It  consists  of  a  piece  of  wood  or  celluloid  that  will 
not  quite  fill  up  the  space  inside  of  a  triangle.  With  the  middle 
finger  of  the  left  hand  on  the  piece  of  wood,  and  the  index 
finger  on  the  triangle,  one  is  slipped  while  the  other  is  held, 
thus  "walking"  in  steps  of  equal  length. 

If  a  piece  of  wood  is  not  available,  should  the  draftsman 
be  so  fortunate  as  to  possess  a  few  coins,  he  can  make  use  of 
the  same  principle  and  get  the  right  spacing  by  some  such 
combination  as  shown  in  Fig.  10. 


KINKS  AND  SHORT  CUTS 


31 


7.  Fig.  11  shows  an  adjustable  three-piece  section  liner. 
It  can  easily  be  whittled  out  of  hard  wood.  The  author  has 
used  one  made  by  himself  for  ten  years.  To  set  it  to  a  given 
space  proceed  as  follows : 

Draw  two  lines  a  and  b  at 
the  proper  distance  apart. 

Set  the  triangle  on  b. 

Slip  clamp  c  to  the  left  to 
loosen  the  device,  and  spread  d 
and  e  as  far  apart  as  the  triangle 
will  permit. 

Hold  d  fast,  slip  the  triangle 
to  a,  move  c  to  the  right,  and 
the  liner  is  set. 


FIG.  10. 


8.  To  Rectify  an  Arc.  It  is  theoretically  as  well  as 
practically  impossible  to  lay  off  the  exact  length  of  an  arc  on  a 
straight  line,  or,  a  straight  line  on  an  arc.  However,  the 


32  KINKS  AND  SHORT  CUTS 

degree  of  accuracy  attainable  is  not  limited  by  theoretical  con- 
siderations, but,  as  in  all  other  measurements,  by  the  accuracy 
of  instruments  and  the  acuteness  of  sight  and  touch. 

The  method  generally  used  is  that  illus- 
trated in  Fig.  12,  in  which  TO  is  the  given 
arc  and  TQ  a  tangent  at  T. 

Starting  at  O  small  spaces  are  stepped 
off  with  the  dividers  until  a  point  P,  less 
than  one  step  from  T,  is  reached,  from 
which  point  the  same  number  of  steps  are 
taken  on  TQ. 

Theoretically,  by  this  method  the  error  may  be  made  less 
than  any  assignable  value  by  making  the  steps  sufficiently 
short,  since  the  shorter  the  chord  the  nearer  it  approaches  in 
length  to  the  arc  which  it  subtends. 

Practically,  an  error  of  measurement  is  made  with  each  step 
and  the  resultant  error  is  the  difference  between  the  algebraic 
sum  of  errors  made  in  stepping  off  the  arc,  less  the  algebraic 
sum  of  errors  made  in  stepping  off  the  straight  line.  Hence, 
the  probable  sum  total  of  errors  in  measurement  varies  directly 
with  the  number  of  steps. 

There  are  a  number  of  approximate  methods  of  rectifying 
an  arc.  The  following  is  a  method  devised  by  the  author: 

9.  The  Rectifier.  On  a  stiff  piece  of  paper  a  circle  of 
some  commensurate  diameter  is  drawn,  Fig.  13.  AB  is  drawn 
tangent  to  the  circle  and  B  is  carefully  located  by  an  accurate 
scale  so  that  AB  equals  the  computed  length  of  the  semi-, 
circumference  AC. 

AB  and  AC  are  divided  and  subdivided  into  the  same  number 
of  parts.  Tangents  are  drawn  to  the  circle  at  each  division  and 
the  proper  length  taken  from  AB  and  laid  off  on  each  to  plot 
the  involute  CB. 

A  templet  is  cut  out  as  shown  in  Fig.  13. 


KINKS  AND  SHORT  CUTS 


33 


The  following  explains  the  use  of  the  rectifier. 
To  rectify  on  PQ  the  arc  TO.*     Fig.  14. 


FIG.  13. 


FIG.  14. 


Place  the  pricker  point  at  K,  Fig.  15,  and  slip  the  notch  of 
the  templet  against  it. 


FIG.  15. 


Turn  until  R  of  the  templet  falls  on  the  line  KL. 

Draw  AB  by  ruling  along  AB  of  the  templet. 

Turn  the  templet  about  K  until  R  falls  on  KO,  Fig.  16, 
and  mark  where  the  involute  intersects  AB  at  N. 

Draw  the  line  KN  until  it  intersects  TQ  at  Q.  TQ  is  the 
arc  TO  rectified  as  required. 

Fig.  17  is  an  enlarged  view  of  the  notch  K. 


*  The  same  method  is  used  when  the  radius  of  the  arc  is  less  than 
the  radius  of  the  templet  circle.  \ 


34  KINKS  AND  SHORT  CUTS 

10.  Plotting  Trochoidal  Curves.      In  Fig.  18,  A,  B,  C, 
and  D  are  points  fixed  to  M.     Let  it  be  required  to  plot  the 
locus  of  each  when  M  is  rolled  on  N. 

These  loci  may  be  drawn  by  plotting  the  loci  of  any  two 
points  fixed  to  M. 

Let  the  locus  of  A  and  D  be  chosen. 

The  locus  of  A  is  the  circle  AA'  with  E  as  center. 

When  A  is  at  A'  D  will  lie  on  an  arc  of  a  circle  with 
center  A'  and  radius  equal  to  AD. 

D  will  also  lie  on  the  line  A'C'  extended  where  Rect.  arc 
FC'=Rect.  arc  FC. 

Any  number  of  positions  of  A  and  D  may  be  thus  obtained. 

The  accuracy  of  this  method  depends  on  the  accuracy  with 
which  the  locus  of  C,  which  is  an  epicycloid,  is  plotted. 

An  easier  and  more  accurate  method  is  to  plot  the  locus  of 
A  and  E  as  follows: 

With  E,  Fig.  19,  as  center,  draw  an  arc  thru  A,  and  with  A 
as  center  draw  an  arc  thru  E. 

angle  AEK      AC 

Lay  off  K  and  L  so  that   —  —  =  - — 

angle  EAL       CE 

Divide  AK  and  EL  into  the  same  number  of  equal  parts 
(1,  2,  3,  etc.,  on  AK,  and  1',  2',  3',  etc.,  on  EL). 

With  1,  2,  3,  etc.,  as  centers  strike  arcs  thru  E. 

With  E  as  center  strike  arcs  thru  I',  2',  3',  etc. 

The  intersections  of  these  arcs  at  a,  b,  c,  etc.,  are  points  on 
the  locus  of  E  when  E  is  a  point  on  M,  and  M  is  rolled  on  N.. 

11.  The  Trochoidal  Templet.     The  last  method  above 
was  used  in  plotting  the  trochoidal  templet,  Fig.  20. 

The  points  in  the  imaginary  circle  U  is  the  locus  of  A  fixed 
to  M  when  M  rolls  on  N. 

V  is  the  imaginary  circle  described  by  E  fixed  to  N  when 
N  rolls  on  M. 

W  is  the  locus  of  E  fixed  to  M  when  M  rolls  on  N. 

X  is  the  locus  of  A  fixed  to  N  when  N  rolls  on  M. 


KINKS  AND  SHORT  CUTS 


35 


Any  trochoidal  curves  of  M  and  N  may  be  plotted  as  follows : 

Place  the  trochoidal  templet  over  the  paper  on  which  the 
curves  are  to  be  plotted. 

Make  a  templet  similar  to  that  shown  in  Fig.  21  where 
FG=AE. 

Mark  on  this  templet  the  points  to  be  plotted. 


FIG.  16. 

If  M  is  to  roll  on  N  place  the  pricker  point  successively  at 
1,  2,  3,  etc.,  of  U  in  Fig.  20  and  slip  F  of  Fig.  21  against  it 
while  the  sharp  end  is  at  the  corresponding  point  of  W. 

Prick  thru  the  points  on  the  notched  templet 
in  each  position.  This  will  mark  points  of  the 
required  loci  on  the  paper  underneath. 

In  making  Fig.  23  the  points  on  U  and  the 
locus  of  a  point  on  the  circumference  of  M, 
Fig.  19,  were  pricked  thru  to  the  paper  on 
which  the  drawing  was  to  be  made. 

A  templet  shown  in  Fig.  22  was  then  made 
to    follow    these    points    as   just   described,    and 
with  a  soft  pencil,  the  tooth  curve  on  the  templet 
was  drawn  in  each  of  its  positions. 


FIG.  17. 


36 


KINKS  AND  SHORT  CUTS 


KINKS  AND  SHORT  CUTS 


37 


38 


KINKS  AND  SHORT  CUTS 


FIG.   20. 


KINKS  AND  SHORT  CUTS 


39 


Problems.  1.  (a)  Construct  a  trochoidal  templet  for 
pitch  circles  6  and  8  inches  diameter  respectively.  See  Arts. 
10  and  11.  (b)  Plot  a  series  of  trochoidal  curves  using  a 
moving  templet  with  points  located  as  in  Fig.  21.  (c)  Make 


FIG.  22. 


a  templet  of  a  gear  tooth  for  the  6-inch  pitch  circle  and  find  its 
conjugate  tooth  on   the  8-inch  circle.     See   Figs.   22  and  23. 

2.  (a)      Draw  Brown  and  Sharpe  Standard  involute  teeth 
of  two  meshing  gears.     Diam.  pitch=2;  12  and  16  teeth. 

3.  Make   a   layout   drawing   of   two   bevel   gears.      Diam. 
pitch=3,  30  and  38  teeth,  axes  intersecting  at  an  angle  of  75°. 
Dimension  by  scaling.     See  Plate  22,  Page  44. 


40 


KINKS  AND  SHORT  CUTS 


4.     Compute    the    necessary    dimensions    of    Prob.    3,    using 
instructions  on  Plates  19,  20,  and  21,  Pages  41,  42  and  43. 


TIG.  23. 


KINKS  AND  SHORT  CUTS 


41 


BEVEL  GEAR  FORMULAE 
AXES  3O° 

QUANTITY 

"•» 

vj 

|?       FORMULAE 
52 

Diametral  Pitch 

\    These  are  defer 
mined  by  the 
fr  L  vwrfring  condi- 
tions of  the 
c\  gears. 

Number  0f  Teeth  in  Gear  1 

II                         II                       H                 II                  II               ^ 

Addendum 

d=±  (B.8cS.$t'd) 

&7 

Pedendum 

JJS7     fa  a   c     c^l 

&  -                 \U.  (J£<J.   (J  1  cij 

Constant 

f=Mb*+c* 

Pitch  Angle  0f  6e#r  1 

9=  T"n:'T 

2 

h=   90°-  g 

Addendum  Angle 

•  _  a*d       <*{:O 

'     3./4/6*fXj°U 

Pedendum  Ang/e 

J-  >^i 

Addendum  Increase 
of  Gearl 

A=  ^xc 

Addendum  Increase 
of  Gear  £ 

m=4*b 

PLATE  19. 


42 


KINKS  AND  SHORT  CUTS 


BEVEL  GEARS 
CALCULATED  DIMENSIONS 


PLATE  20. 


KINKS  AND  SHORT  CUTS 


43 


BEVEL  GEARS 
CALCULATED  DIMENSIONS 


6 ear  2 


Number  of  Teeth  in  Gear  1  =b 
Number  of  Teeth  in  GearB  =B 
Ang/e  of  Axes  =/3 

Number  of  Teeth  in  the  & 

Imaginary  6ear  2.  ~tan@ 

For  other  quantities  substitute  in  PLATE  19 


PLATE  21 


44 


KINKS  AND  SHORT  CUTS 


BEVEL 
OBTAINING  THE  MACHINED 
DIMENSIONS  BY  LAYOUT 

REQUIRED:  A  layout  for  cutting  the  teeth  of  two 
bevel  gears  of  3  Pitch,  12  and  18  Teeth  respective- 
ly,  /£  Face,   Axes  perpendicular 


Dedendum 


Addendum^*, 


Pitch  Diam. 


Pitch  P/am.  = 


Compare  with  PLATE. 

PLATE  22. 


Books    on    the    Manual    Arts 


PROBLEMS  IN  MECHANICAL  DRAWING. 

By  Charles  A.  Bennett.    With  drawings  made  by 
Fred  D.  Crawshaw. 

This  book  consists  of  80  plates  and  a  few  explanatory  notes,  and  is  bound  loose 
leaf,  in  board  covers  with  brass  fasteners.  Its  purpose  is  to  furnish  teachers  of 
classes  beginning  mechanical  drawing  with  a  large  number  of  simple,  practical 
problems.  These  have  been  selected  with  reference  to  the  formation  of  good  habits 
in  technique,  the  interest  of  the  pupils,  and  the  subjects  generally  included  in  a 
grammar  and  first-year  high  school  course.  Each  problem  given  is  unsolved  and 
therefore  in  proper  form  to  hand  to  the  pupil  for  solution.  Price,  $1.00. 

MECHANICAL  DRAFTING.     By  H.  W.  Miller. 

A  new  departure  among  textbooks  on  mechanical  drawing.  It  is  intended  to 
supplement  the  work  of  the  instructor  in  such  a  way  as  to  reduce  lecture  work  to 
a  minimum.  It  is  written  about  a  flexible  course  but  may  be  used  equally  well  with 
any  course.  The  book  abounds  in  illustrations,  both  line  drawings  and  half-tones, 
It  shows  a  wise  selection  of  material,  a  keen  insight  into  the  work  of  the  draftsman 
and  a  thoro  knowledge  of  the  principles  and  methods  of  teaching.  Above  all  it  is  a 
practical  treatment  of  subject  matter  and  a  students'  text  easily  adaptable  to  varied 
schools  and  conditions.  Contains  219  pages  and  225  illustrations  and  is  bound  in 
black  flexible  leather,  pocket  book  size.  Price,  $1.50. 

DESCRIPTIVE  GEOMETRY.     By  H.  W.  Miller. 

A  successful  textbook  that  is  at  once  clear  and  terse  in  expression,  complete  in 
treatment  and  logical  in  arrangement.  It  treats  of  Point,  Line  and  Plane,  Inter- 
sections and  Developments,  Shades  and  Shadows  and  Linear  Perspective.  It  con- 
tains about  1,000  graphic  problems  and  is  bound  in  leather,  pocket  book  size. 
Price,  $1.50. 

THE     WASH     METHOD     OF     HANDLING     WATER- 
COLOUR.     By  Frank  Forrest  Frederick. 

A  brief,  clear,  comprehensive  text  printed  in  sepia  and  illustrated  with  wash 
drawings  and  a  water-color  painting  by  the  author.  Price,  50  cents. 

SIMPLIFIED  MECHANICAL  PERSPECTIVE. 

By  Frank  Forrest  Frederick. 

A  book  of  simple  problems  covering  the  essentials  of  mechanical  perspective. 
It  is  planned  for  pupils  of  high  school  age  who  have  already  received  some  elementary 
training  in  mechanical  drawing.  It  is  simple,  direct  and  practical.  Price,  75  cents. 

HANDWORK  IN  WOOD.     By  William  Noyes. 

A  handbook  for  teachers  and  a  textbook  for  normal  school  and  college  students. 
A  comprehensive  and  scholarly  treatise,  covering  logging,  sawmilling,  seasoning 
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304  illustrations — excellent  pen  drawings  and  many  photographs.  Price,  $2.00. 


WOOD  AND  FOREST.     By  William  Noyes. 


Chotograpns    ana    micropnotograpns    01    sections.      Domains    a    general    oionography    01 
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Noyes  and   photographs  by   the  author.      309    pages.      Price,   $3.00. 

Published  by  the  Manual  Arts  Press     -     -     Peoria,  111. 


THE  DRAFTING  ROOM  SERIES 


By  FREDERICK  H.  EVANS,  M.  E. 

Assiitant  Professor  of  Manual  Arts 
Bradley  Polytechnic  Institute,  Peoria,  Illinois 


A    MODERN    AND    EXTREMELY    PRACTICAL    TREAT- 
MENT OF  COMMERCIAL  DRAFTING 
IN  CARD  INDEX  FORM 


P*HIS  unique  work  is  the  result  of  a  new  analysis  of  the  processes  of  com- 
•*•  mercial  drafting  by  a  practical  draftsman,  engineer  and  teacher.  The 
book  points  out  the  quickest,  most  exact  and  most  practical  method  of  work  to 
obtain  the  desired  results. 

Altho  essentially  a  textbook  containing  problems,  explanations,  information 
and  necessary  data,  it  is  an  inspiration  to  collect  from  time  to  time  additional 
material  for  ready  reference  and  constant  use.  The  scope  of  the  book  ranges 
from  tools  and  their  uses  to  gears  and  moving  parts.  In  the  selection  of  prob- 
lems the  author  has  refrained  from  choosing  merely  difficult  and  time-consuming 
problems  and  has  presented  problems  selected  because  they  definitely  present 
certain  principles.  The  aim  has  been  to  develop  skill  not  thru  repetition  but 
thru  understanding;  not  so  much  results,  as  methods  and  processes.  In  all,  it 
is  a  textbook  worthy  in  form,  in  content  and  in  adaptability  to  be  considered 
for  adoption  by  every  teacher  of  machine  drawing. 

It  is  divided  into  three  parts.  The  first  part,  READING  MACHINE 
DRAWINGS,  is  intended  for  beginners'  use.  It  is  designed  to  teach  reading 
of  drawings,  and  requires  the  use  of  only  an  ordinary  lead  pencil  and  a  pad  of 
cross-lined  paper.  It  consists  of  a  20-page  pamphlet  and  17  cards. 

The  second  part,  MACHINE  DRAFTING,  contains  practical  informa- 
tion on  the  common  instruments,  materials  and  tools  of  the  draftsman  and  is  a 
treatment  at  length  of  the  practical  work  of  the  designer,  detailer,  checker  and 
tracer,  and  a  discussion  of  drafting  room  records,  systems,  etc.  It  contains  a 
general  treatment  of  "The  Drafting  Room",  '"Detailing",  "Checking", 


PEORIA,  ILL. 


THE  DRAFTING  ROOM  SERIES 


By  FREDERICK  H.  EVANS,  M.  E. 

Assistant  Professor  of  Manual  Arts 
Bradley  Polytechnic  Institute,  Peoria,  Illinois 


A  SUCCESSFUL  TEXTBOOK  FOR  VOCATIONAL 

SCHOOL,  EVENING  SCHOOL,  TECHNICAL 

SCHOOL,  AND   ENGINEERING 

STUDENTS 


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I.N  USE 

"Tracing",  "The  Geometry  of  Drafting"  and  "Kinks  and  Short  Cuts".  It 
presents  problems  in  drawing,  detailing,  assembling,  checking,  tracing,  etc., 
and  contains  9  pages  of  sketches  and  notes  on  the  fundamental  mechanical  prin- 
ciples and  their  common  applications  in  machinery.  It  consists  of  a  48-page 
pamphlet  and  44  cards. 

The  third  part,  INTERFERENCE  OF  MOVING  PARTS  AND 
TOOTH  GEARS,  contains  an  entirely  new  presentation  of  gears,  avoiding 
technical  language  and  difficult  formulae,  yet  going  to  the  bottom  of  the  matter 
with  perfect  clearness.  The  subjects  treated  of  are  "Interference  of  Moving 
Parts",  "Transmission  of  Motion  by  Moving  Contact",  "Tooth  Gears"  and 
the  "General  Principles  of  Conjugate  Curves".  It  presents  problems  in  calcu- 
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This  part  consists  of  a  40-page  pamphlet  and  21  cards. 

The  series  complete  consists  of  three  pamphlets  and  fifty-four  cards,  no 
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PRICE,  COMPLETE,  $2.00 

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ESSENTIALS  OF  WOODWORKING.     By  Ira  S.  Griffith. 

A  textbook  written  especially  for  the  use  of  grammar  and  high  school  students. 
A  clear  and  comprehensive  treatment  of  woodworking  tools,  materials,  and  processes, 
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is  illustrated  with  photographs  and  numerous  pen  drawings.  Price,  $1.00. 

CORRELATED     COURSES     IN     WOODWORK     AND 
MECHANICAL  DRAWING.   By  Ira  S.  Griffith. 

This  book  is  designed  to  meet  the  every-day  need  of  the  teacher  of  woodworking 
and  mechanical  drawing  for  reliable  information  concerning  organization  of  courses, 
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PROJECTS     FOR     BEGINNING     WOODWORK     AND 
MECHANICAL  DRAWING.   By  Ira  S.  Griffith. 

A  work  book  for  the  use  of  students  in  grammar  grade  classes.  It  consists  of 
working  drawings  and  working  directions.  The  projects  are  such  as  have  proven 
of  exceptional  service  where  woodworking  and  mechanical  drawing  are  taught  in  a 
thoro,  systematic  manner  in  the  seventh  and  eighth  grades.  The  aim  has  been  to 
provide  successful  rather  than  unique  problems.  The  50  projects  in  the  book  have 
been  selected  and  organized  with  the  constant  aim  of  securing  the  highest  educational 
results.  The  book  is  especially  suited  for  use  in  connection  with  "Essentials  of 
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ADVANCED  PROJECTS  IN  WOODWORK. 

By  Ira  S.  Griffith. 

This  book  is  similar  to  "Projects  for  Beginning  Woodwork  and  Mechanical 
Drawing",  but  is  suited  to  high  school  needs.  It  consists  of  fifty  plates  of  problems 
and  accompanying  notes.  It  is  essentially  a  collection  of  problems  in  furniture 


PROBLEMS  IN  FURNITURE  MAKING. 
By  Fred  D.  Crawshaw. 

This  book,  revised  and  enlarged,  consists  of  43  plates  of  working  drawings 
suitable  for  use  in  grammar  and  high  schools,  and  36  pages  of  text,  including 
chapters  on  design,  construction  and  finishes,  and  notes  on  the  problems.  Price,  $1.00. 

PROBLEMS  IN  WOOD-TURNING. 

By  Fred  D.  Crawshaw. 

In  the  first  place  this  is  a  book  of  problems — 25  plates  covering  spindle,  face- 
plate, and  chuck  turning.  In  the  second  place  it  is  a  textbook  on  the  science  and 
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basis  for  the  cuts  used  in  turning.  In  the  third  place  it  is  a  helpful  discussion  of 
the  principles  of  design  as  applied  to  objects  turned  in  wood.  It  is  a  clear,  practical 
and  suggestive  book  on  wood-turning.  Price,  80  cents.  Board  covers,  $1.00. 

WOOD  PATTERN-MAKING.     By  Horace  T.  Purfield. 

This  book  was  written  expressly  for  use  as  a  textbook  for  high  school,  trade 
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A  catalog,  listing  and  describing  over  300  books  on  the  Manual 
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